scholarly journals Birth Rate Effects on an Age-Structured Predator-Prey Model with Cannibalism in the Prey

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Francisco J. Solis ◽  
Roberto A. Ku-Carrillo
Keyword(s):  
Birth Rate ◽  
Birth Rates ◽  
Differential Systems ◽  
Predator Prey ◽  
Predator Prey Model ◽  
New Birth ◽  
Rate Effects ◽  
Stable Equilibria ◽  
Predator Prey Models ◽  
Age Structured

We develop a family of predator-prey models with age structure and cannibalism in the prey population. It consists of systems ofmordinary differential equations, wheremis a parameter associated with new proposed prey birth rates. We discuss how these new birth rates give the required flexibility to produce differential systems with well-behaved solutions. The main feature required in these models is the coexistence among the involved species, which translates mathematically into stable equilibria and periodic solutions. The search for such characteristics is based on heuristic predation functions that account for cannibalism in the prey.

scholarly journals Qualitative Analysis of a Resource Management Model and Its Application to the Past and Future of Endangered Whale Populations

2021 ◽  
Vol 14 (1) ◽  
pp. 1-18
Author(s):  
Ledder Ledder
Keyword(s):  
Rapid Decline ◽  
Predator Population ◽  
Population Declines ◽  
Phase Line ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Intermediate Consumption ◽  
Stable Equilibria ◽  
Predator Prey Models ◽  
Type 3

Observed whale dynamics show drastic historical population declines, some of which have not been reversed in spite of restrictions on harvesting. This phenomenon is not explained by traditional predator prey models, but we can do better by using models that incorporate more sophisticated assumptions about consumer-resource interaction. To that end, we derive the Holling type 3 consumption rate model and use it in a one-variable differential equation obtained by treating the predator population in a predator-prey model as a parameter rather than a dynamic variable. The resulting model produces dynamics in which low and high consumption levels lead to single high and low-level stable resource equilibria, respectively, while intermediate consumption levels result in both high and low stable equilibria. The phase line analysis is made more transparent by applying a particular structure to the function that gives the derivative in terms of the state. By positing a consumption level that starts low, gradually increases through technological change and human population growth, and decreases as a result of public policy, we are able to tell a story that explains the unexpectedly rapid decline of some resources, such as whales, followed by limited recovery in response to conservation. The analysis also offers guidelines for how to establish sustainable harvesting for restored populations. We include a bifurcation analysis and suggestions for how to teach the material with three different levels of focus on the modeling aspect of the study.

Bifurcation of Codimension 3 in a Predator–Prey System of Leslie Type with Simplified Holling Type IV Functional Response

2016 ◽  
Vol 26 (02) ◽  
pp. 1650034 ◽  
Author(s):  
Jicai Huang ◽  
Xiaojing Xia ◽  
Xinan Zhang ◽  
Shigui Ruan
Keyword(s):  
Functional Response ◽  
Limit Cycle ◽  
Stable Limit Cycle ◽  
Positive Equilibrium ◽  
Stable Limit ◽  
Homoclinic Loop ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Stable Equilibria

It was shown in [Li & Xiao, 2007] that in a predator–prey model of Leslie type with simplified Holling type IV functional response some complex bifurcations can occur simultaneously for some values of parameters, such as codimension 1 subcritical Hopf bifurcation and codimension 2 Bogdanov–Takens bifurcation. In this paper, we show that for the same model there exists a unique degenerate positive equilibrium which is a degenerate Bogdanov–Takens singularity (focus case) of codimension 3 for other values of parameters. We prove that the model exhibits degenerate focus type Bogdanov–Takens bifurcation of codimension 3 around the unique degenerate positive equilibrium. Numerical simulations, including the coexistence of three hyperbolic positive equilibria, two limit cycles, bistability states (one stable equilibrium and one stable limit cycle, or two stable equilibria), tristability states (two stable equilibria and one stable limit cycle), a stable limit cycle enclosing a homoclinic loop, a homoclinic loop enclosing an unstable limit cycle, or a stable limit cycle enclosing three unstable hyperbolic positive equilibria for various parameter values, confirm the theoretical results.

Pattern Formation Controlled by External Forcing in a Spatial Harvesting Predator-Prey Model

2011 ◽  
pp. 214-221
Author(s):  
Feng Rao
Keyword(s):  
Reaction Diffusion ◽  
Euler Method ◽  
External Forcing ◽  
Periodic Forcing ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Flux Boundary Conditions ◽  
Predator Prey Models ◽  
The Stability

Predator–prey models in ecology serve a variety of purposes, which range from illustrating a scientific concept to representing a complex natural phenomenon. Due to the complexity and variability of the environment, the dynamic behavior obtained from existing predator–prey models often deviates from reality. Many factors remain to be considered, such as external forcing, harvesting and so on. In this chapter, we study a spatial version of the Ivlev-type predator-prey model that includes reaction-diffusion, external periodic forcing, and constant harvesting rate on prey. Using this model, we study how external periodic forcing affects the stability of predator-prey coexistence equilibrium. The results of spatial pattern analysis of the Ivlev-type predator-prey model with zero-flux boundary conditions, based on the Euler method and via numerical simulations in MATLAB, show that the model generates rich dynamics. Our results reveal that modeling by reaction-diffusion equations with external periodic forcing and nonzero constant prey harvesting could be used to make general predictions regarding predator-prey equilibrium,which may be used to guide management practice, and to provide a basis for the development of statistical tools and testable hypotheses.

A linear birth-and-death predator–prey process

1995 ◽  
Vol 32 (01) ◽  
pp. 274-277
Author(s):  
John Coffey
Keyword(s):  
Death Rate ◽  
Birth Rate ◽  
Prey Population ◽  
Death Process ◽  
Predator Population ◽  
Birth And Death Process ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

A new stochastic predator-prey model is introduced. The predator population X(t) is described by a linear birth-and-death process with birth rate λ 1 X and death rate μ 1 X. The prey population Y(t) is described by a linear birth-and-death process in which the birth rate is λ 2 Y and the death rate is . It is proven that and iff

Extinction probabilities in predator–prey models

1986 ◽  
Vol 23 (01) ◽  
pp. 1-13
Author(s):  
S. E. Hitchcock
Keyword(s):  
Stochastic Models ◽  
Birth Rate ◽  
Exact Expression ◽  
Initial Population ◽  
Predator Prey ◽  
Extinction Probabilities ◽  
Population Sizes ◽  
The Mean ◽  
Predator Prey Models ◽  
Two Populations

Two stochastic models are developed for the predator-prey process. In each case it is shown that ultimate extinction of one of the two populations is certain to occur in finite time. For each model an exact expression is derived for the probability that the predators eventually become extinct when the prey birth rate is 0. These probabilities are used to derive power series approximations to extinction probabilities when the prey birth rate is not 0. On comparison with values obtained by numerical analysis the approximations are shown to be very satisfactory when initial population sizes and prey birth rate are all small. An approximation to the mean number of changes before extinction occurs is also obtained for one of the models.

scholarly journals Theoretical Studies on the Effects of Dispersal Corridors on the Permanence of Discrete Predator-Prey Models in Patchy Environment

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
Chunqing Wu ◽  
Shengming Fan ◽  
Patricia J. Y. Wong
Keyword(s):  
Sufficient Conditions ◽  
Theoretical Studies ◽  
Patchy Environment ◽  
Predator Prey ◽  
Numerical Examples ◽  
Predator Prey Model ◽  
Prey Model ◽  
Dispersal Corridors ◽  
Predator Prey Models ◽  
Theoretical Results

We study two discrete predator-prey models in patchy environment, one without dispersal corridors and one with dispersal corridors. Dispersal corridors are passes that allow the migration of species from one patch to another and their existence may influence the permanence of the model. We will offer sufficient conditions to guarantee the permanence of the two predator-prey models. By comparing the two permanence criteria, we discuss the effects of dispersal corridors on the permanence of the predator-prey model. It is found that the dispersion of the prey from one patch to another is helpful to the permanence of the prey if the population growth of the prey is density dependent; however, this dispersion of the prey could be disadvantageous or advantageous to the permanence of the predator. Five numerical examples are presented to confirm the theoretical results obtained and to illustrate the effects of dispersal corridors on the permanence of the predator-prey model.

scholarly journals Coexistence for an Almost Periodic Predator-Prey Model with Intermittent Predation Driven by Discontinuous Prey Dispersal

2017 ◽  
Vol 2017 ◽  
pp. 1-15
Author(s):  
Yantao Luo ◽  
Long Zhang ◽  
Zhidong Teng ◽  
Tingting Zheng
Keyword(s):  
Growth Rates ◽  
Prey Species ◽  
Impulsive Differential Equations ◽  
Comparison Theorems ◽  
Almost Periodic ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Predator Prey Models

An almost periodic predator-prey model with intermittent predation and prey discontinuous dispersal is studied in this paper, which differs from the classical continuous and impulsive dispersal predator-prey models. The intermittent predation behavior of the predator species only happens in the channels between two patches where the discontinuous migration movement of the prey species occurs. Using analytic approaches and comparison theorems of the impulsive differential equations, sufficient criteria on the boundedness, permanence, and coexistence for this system are established. Finally, numerical simulations demonstrate that, for an intermittent predator-prey model, both the intermittent predation and intrinsic growth rates of the prey and predator species can greatly impact the permanence, extinction, and coexistence of the population.

FINAL STATE OF ECOSYSTEM CONTAINING GRASS, SHEEP AND WOLVES WITH AGING

2005 ◽  
Vol 16 (01) ◽  
pp. 177-190 ◽  
Author(s):  
MINGFENG HE ◽  
QIU-HUI PAN ◽  
SHUANG WANG
Keyword(s):  
Cellular Automata ◽  
Energy Supply ◽  
Birth Rate ◽  
Reproduction Number ◽  
Final States ◽  
Final State ◽  
Cellular Automata Model ◽  
Predator Prey ◽  
Prey System ◽  
Age Structured

This paper describes a cellular automata model containing movable wolves, sheep and reproducible grass. Each wolf or sheep is characterized by Penna bitstrings. In addition, we introduce the energy rule and the predator–prey mechanism for wolf and sheep. With considering age-structured, genetic strings, minimum reproduction age, cycle of the reproduction, number of offspring, we get three possible states of a predator–prey system: the coexisting one with predators and prey, the absorbing one with prey only, and the empty one where no animal survived. In this paper, we mainly discuss the effect of the number of poor genes, the energy supply (food), the minimum reproduction age, the reproductive cycle and the birth rate on the above three possible final states.

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